Why Lognormal Returns Do Not Give a Simple VaR for Long-Short Portfolios

# How can I estimate value-at-risk of a long/short portfolio without making simplifying assumptions?


# How can I estimate value-at-risk of a long/short portfolio without making simplifying assumptions?












I have had a couple of long-standing questions about the mathematics behind a simple "vanilla" parametric VaR calculation and I'm hoping someone could clear up my confusion. Most likely I am misunderstanding something at a fundamental level so I apologize in advance.

My main question is how to actually perform a "normal" linear VaR given some historical asset prices and some assumed positions, if we assume normality of the log-returns, but specifically do not assume normality of the percentage-returns. A lot of texts I have read seem to equate the two, but I am wondering if there is a way to get to the VaR without treating the two assumptions as interchangeable.

For simplicity, assume 1-day horizon, and no interest rates. In Carol Alexander's Market Risk Analysis series, she essentially defines the VaR as:

$VaR_{\alpha} = -r_{t} P_t$

where $r_t$ is the value such that $P(R_t < r_t) = \alpha$, and where $R_t$ is the random variable denoting the X-day (in this case, 1-day) percentage return of the portfolio between times $t+1$ and $t$.

Further, assume that the 1-day log-returns of the portfolio are denoted by $\hat{R}_{t}$.

Formally, $R_t = \frac{P_{t+1}}{P_t} - 1$ and $\hat{R_t} = ln(\frac{P_{t+1}}{P_t})$ where $P_t$ denotes the portfolio value at time $t$.

This implies the following relationships:

$\hat{R}_{t} = ln(R_t+1)$ and $R_t = e^{\hat{R}_t}-1$

In Carol Alexander's Market Risk Analysis series, she demonstrates that if you assume the percentage return is distributed normally, you can derive: $VaR_{\alpha} = (\Phi^{-1}(1-\alpha)\sigma-\mu) P_t$

However, I am interested in when the log-return is assumed to be distributed normally, but not necessarily the percentage return.

Let $\hat{R}_t \sim N(\mu, \sigma^2)$ where $\mu$ and $\sigma^2$ are the mean and variance of the distribution, respectively.

Then $P(R_t < r_t) = \alpha \implies P(e^{\hat{R}_t}-1 < r_t) = \alpha \implies P(\hat{R}_t < ln(r_t + 1)) = \alpha $

Doing a Z transform,

$P(Z < \frac{ln(r_t+1)-\mu}{\sigma}) = \alpha \implies \frac{ln(r_t+1)-\mu}{\sigma} = \Phi^{-1}(\alpha) \implies r_t = e^{\Phi^{-1}(\alpha)\sigma + \mu}-1$

Thus $VaR_{\alpha} = - (e^{\Phi^{-1}(\alpha)\sigma + \mu}-1) P_t$. I have done the above computation without any reference text, so I apologize in advance if I've made any mistakes, and ask that you correct me if so.

My question is the following: for use in the above formula, I wish to estimate the portfolio log-return's $\mu$ and $\sigma$ from history. Usually I can do this by valuing the portfolio in the past on each day, taking the log-returns between the valuations, and then estimating the parameters from that series. However, what if I have a portfolio such that the portfolio valuations go from positive to negative on some days? Then I cannot estimate the mean or standard deviation from the series because there are undefined values.

For example, suppose I have a portfolio which is long 50 shares of MSFT and short 90 shares of AAPL, from the beginning of 2019 to current. Marking the portfolio using the close price, you can see that this portfolio occasionally goes negative and hence leads to an undefined log-return.

One suggestion I have seen is the "VCV approach" which, if I am understanding correctly, is to take the log-returns of the individual asset prices, and then calculate the log-return of the portfolio from the weighted average of those individual log-return parameters.

Essentially, this asserts that:

$VaR(\hat{R}) = Var(aX+bY) = a^2VaR(X)+b^2VaR(Y) + 2abCov(X,Y)$ where a and b would be the notional values (price * quantity) invested in the two stocks, respectively, and X and Y would be the random variables denoting the log returns of the two stocks, respectively.

Using this approach gets me to a value; however, I'm confused as to why we are actually able to do this, because it was my understanding that being able to split out the return like this was a property of the percentage return but was not guaranteed to be true for the log-return. (Aka, $R_t = aX + bY$ for % returns of the stocks, but $\hat{R_t} \neq aX + bY$ for log-returns of the stocks).

My other idea was to do some kind of affine transformation to the data to remove negative values, but wasn't sure how to implement this. I found that changing the exact value I "shifted" the data by would change my final VaR result, which seems unintuitive.

If anyone could either link me to a piece of literature which addresses this, correct my math if I've made a foolish error, or explain to me the proper procedure, I'd be really grateful.

Also: if the answer is simply, "don't assume a lognormal distribution when there are negative values," I'd accept that, but would just want to confirm that is the proper way I should be thinking about it.